In Adams' blue book (page 54) we have a map in the homotopy category of ring spectra $f: MU \rightarrow K$ where $K$ is complex $K$-theory such that $g_*x^{MU} = (u^K)^{-1}x^K$ where $x^E$ denote complex orientation of a ring spectrum $E$. My question is - does this refine to a map of $E_{\infty}$-ring spectra (or a map of commutative ring spectra in one of these modern categories of spectra)?
In general, are maps $MU \rightarrow E$ induced by complex orientation on $E$ maps of $E_{\infty}$ ring specta?
References/proofs would be hugely appreciated!
A bit late on this, but this type of question is considered by Chadwick and Mandell in http://arxiv.org/abs/1310.3336, although they give conditions on when the map is $E_2$. This suggests that in general the problem is difficult!